Reverse Cuthill–McKee

Cuthill-McKee ordering of matrices

The reverse Cuthill–McKee algorithm gives a sparse matrix ordering that reduces the matrix bandwidth.

ordering [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (2, 1), (1, 2), (2, 2)]
unordered Laplacian matrix
bandwidth: 7
<Compressed Sparse Row sparse array of dtype 'int64'
        with 33 stored elements and shape (9, 9)>
  Coords        Values
  (0, 0)        2
  (0, 1)        -1
  (0, 3)        -1
  (1, 0)        -1
  (1, 1)        3
  (1, 2)        -1
  (1, 4)        -1
  (2, 1)        -1
  (2, 2)        2
  (2, 5)        -1
  (3, 0)        -1
  (3, 3)        3
  (3, 4)        -1
  (3, 6)        -1
  (4, 1)        -1
  (4, 3)        -1
  (4, 4)        4
  (4, 5)        -1
  (4, 7)        -1
  (5, 2)        -1
  (5, 4)        -1
  (5, 5)        3
  (5, 8)        -1
  (6, 3)        -1
  (6, 6)        2
  (6, 7)        -1
  (7, 4)        -1
  (7, 6)        -1
  (7, 7)        3
  (7, 8)        -1
  (8, 5)        -1
  (8, 7)        -1
  (8, 8)        2
low-bandwidth Laplacian matrix
bandwidth: 7
<Compressed Sparse Row sparse array of dtype 'int64'
        with 33 stored elements and shape (9, 9)>
  Coords        Values
  (0, 0)        2
  (0, 1)        -1
  (0, 2)        -1
  (1, 0)        -1
  (1, 1)        3
  (1, 3)        -1
  (1, 4)        -1
  (2, 0)        -1
  (2, 2)        3
  (2, 4)        -1
  (2, 5)        -1
  (3, 1)        -1
  (3, 3)        2
  (3, 6)        -1
  (4, 1)        -1
  (4, 2)        -1
  (4, 4)        4
  (4, 6)        -1
  (4, 7)        -1
  (5, 2)        -1
  (5, 5)        2
  (5, 7)        -1
  (6, 3)        -1
  (6, 4)        -1
  (6, 6)        3
  (6, 8)        -1
  (7, 4)        -1
  (7, 5)        -1
  (7, 7)        3
  (7, 8)        -1
  (8, 6)        -1
  (8, 7)        -1
  (8, 8)        2

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import networkx as nx


# build low-bandwidth matrix
G = nx.grid_2d_graph(3, 3)
rcm = list(nx.utils.reverse_cuthill_mckee_ordering(G))
print("ordering", rcm)

print("unordered Laplacian matrix")
A = nx.laplacian_matrix(G)
x, y = np.nonzero(A)
# print(f"lower bandwidth: {(y - x).max()}")
# print(f"upper bandwidth: {(x - y).max()}")
print(f"bandwidth: {(y - x).max() + (x - y).max() + 1}")
print(A)

B = nx.laplacian_matrix(G, nodelist=rcm)
print("low-bandwidth Laplacian matrix")
x, y = np.nonzero(B)
# print(f"lower bandwidth: {(y - x).max()}")
# print(f"upper bandwidth: {(x - y).max()}")
print(f"bandwidth: {(y - x).max() + (x - y).max() + 1}")
print(B)

sns.heatmap(B.todense(), cbar=False, square=True, linewidths=0.5, annot=True)
plt.show()